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Theorem dcomex 9474
Description: The Axiom of Dependent Choice implies Infinity, the way we have stated it. Thus, we have Inf+AC implies DC and DC implies Inf, but AC does not imply Inf. (Contributed by Mario Carneiro, 25-Jan-2013.)
Assertion
Ref Expression
dcomex ω ∈ V

Proof of Theorem dcomex
Dummy variables 𝑡 𝑠 𝑥 𝑓 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1n0 7732 . . . . . . 7 1𝑜 ≠ ∅
2 df-br 4788 . . . . . . . 8 ((𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) ↔ ⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1𝑜, 1𝑜⟩})
3 elsni 4334 . . . . . . . . 9 (⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1𝑜, 1𝑜⟩} → ⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1𝑜, 1𝑜⟩)
4 fvex 6344 . . . . . . . . . 10 (𝑓𝑛) ∈ V
5 fvex 6344 . . . . . . . . . 10 (𝑓‘suc 𝑛) ∈ V
64, 5opth1 5072 . . . . . . . . 9 (⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1𝑜, 1𝑜⟩ → (𝑓𝑛) = 1𝑜)
73, 6syl 17 . . . . . . . 8 (⟨(𝑓𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1𝑜, 1𝑜⟩} → (𝑓𝑛) = 1𝑜)
82, 7sylbi 207 . . . . . . 7 ((𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → (𝑓𝑛) = 1𝑜)
9 tz6.12i 6357 . . . . . . 7 (1𝑜 ≠ ∅ → ((𝑓𝑛) = 1𝑜𝑛𝑓1𝑜))
101, 8, 9mpsyl 68 . . . . . 6 ((𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → 𝑛𝑓1𝑜)
11 vex 3354 . . . . . . 7 𝑛 ∈ V
12 1oex 7724 . . . . . . 7 1𝑜 ∈ V
1311, 12breldm 5466 . . . . . 6 (𝑛𝑓1𝑜𝑛 ∈ dom 𝑓)
1410, 13syl 17 . . . . 5 ((𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → 𝑛 ∈ dom 𝑓)
1514ralimi 3101 . . . 4 (∀𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
16 dfss3 3741 . . . 4 (ω ⊆ dom 𝑓 ↔ ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
1715, 16sylibr 224 . . 3 (∀𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → ω ⊆ dom 𝑓)
18 vex 3354 . . . . 5 𝑓 ∈ V
1918dmex 7249 . . . 4 dom 𝑓 ∈ V
2019ssex 4937 . . 3 (ω ⊆ dom 𝑓 → ω ∈ V)
2117, 20syl 17 . 2 (∀𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → ω ∈ V)
22 snex 5037 . . 3 {⟨1𝑜, 1𝑜⟩} ∈ V
2312, 12fvsn 6592 . . . . . . . 8 ({⟨1𝑜, 1𝑜⟩}‘1𝑜) = 1𝑜
2412, 12funsn 6081 . . . . . . . . 9 Fun {⟨1𝑜, 1𝑜⟩}
2512snid 4348 . . . . . . . . . 10 1𝑜 ∈ {1𝑜}
2612dmsnop 5750 . . . . . . . . . 10 dom {⟨1𝑜, 1𝑜⟩} = {1𝑜}
2725, 26eleqtrri 2849 . . . . . . . . 9 1𝑜 ∈ dom {⟨1𝑜, 1𝑜⟩}
28 funbrfvb 6381 . . . . . . . . 9 ((Fun {⟨1𝑜, 1𝑜⟩} ∧ 1𝑜 ∈ dom {⟨1𝑜, 1𝑜⟩}) → (({⟨1𝑜, 1𝑜⟩}‘1𝑜) = 1𝑜 ↔ 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜))
2924, 27, 28mp2an 672 . . . . . . . 8 (({⟨1𝑜, 1𝑜⟩}‘1𝑜) = 1𝑜 ↔ 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜)
3023, 29mpbi 220 . . . . . . 7 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜
31 breq12 4792 . . . . . . . 8 ((𝑠 = 1𝑜𝑡 = 1𝑜) → (𝑠{⟨1𝑜, 1𝑜⟩}𝑡 ↔ 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜))
3212, 12, 31spc2ev 3452 . . . . . . 7 (1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜 → ∃𝑠𝑡 𝑠{⟨1𝑜, 1𝑜⟩}𝑡)
3330, 32ax-mp 5 . . . . . 6 𝑠𝑡 𝑠{⟨1𝑜, 1𝑜⟩}𝑡
34 breq 4789 . . . . . . 7 (𝑥 = {⟨1𝑜, 1𝑜⟩} → (𝑠𝑥𝑡𝑠{⟨1𝑜, 1𝑜⟩}𝑡))
35342exbidv 2004 . . . . . 6 (𝑥 = {⟨1𝑜, 1𝑜⟩} → (∃𝑠𝑡 𝑠𝑥𝑡 ↔ ∃𝑠𝑡 𝑠{⟨1𝑜, 1𝑜⟩}𝑡))
3633, 35mpbiri 248 . . . . 5 (𝑥 = {⟨1𝑜, 1𝑜⟩} → ∃𝑠𝑡 𝑠𝑥𝑡)
37 ssid 3773 . . . . . . 7 {1𝑜} ⊆ {1𝑜}
3812rnsnop 5758 . . . . . . 7 ran {⟨1𝑜, 1𝑜⟩} = {1𝑜}
3937, 38, 263sstr4i 3793 . . . . . 6 ran {⟨1𝑜, 1𝑜⟩} ⊆ dom {⟨1𝑜, 1𝑜⟩}
40 rneq 5488 . . . . . . 7 (𝑥 = {⟨1𝑜, 1𝑜⟩} → ran 𝑥 = ran {⟨1𝑜, 1𝑜⟩})
41 dmeq 5461 . . . . . . 7 (𝑥 = {⟨1𝑜, 1𝑜⟩} → dom 𝑥 = dom {⟨1𝑜, 1𝑜⟩})
4240, 41sseq12d 3783 . . . . . 6 (𝑥 = {⟨1𝑜, 1𝑜⟩} → (ran 𝑥 ⊆ dom 𝑥 ↔ ran {⟨1𝑜, 1𝑜⟩} ⊆ dom {⟨1𝑜, 1𝑜⟩}))
4339, 42mpbiri 248 . . . . 5 (𝑥 = {⟨1𝑜, 1𝑜⟩} → ran 𝑥 ⊆ dom 𝑥)
44 pm5.5 350 . . . . 5 ((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → (((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
4536, 43, 44syl2anc 573 . . . 4 (𝑥 = {⟨1𝑜, 1𝑜⟩} → (((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)))
46 breq 4789 . . . . . 6 (𝑥 = {⟨1𝑜, 1𝑜⟩} → ((𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)))
4746ralbidv 3135 . . . . 5 (𝑥 = {⟨1𝑜, 1𝑜⟩} → (∀𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∀𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)))
4847exbidv 2002 . . . 4 (𝑥 = {⟨1𝑜, 1𝑜⟩} → (∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)))
4945, 48bitrd 268 . . 3 (𝑥 = {⟨1𝑜, 1𝑜⟩} → (((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)))
50 ax-dc 9473 . . 3 ((∃𝑠𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓𝑛 ∈ ω (𝑓𝑛)𝑥(𝑓‘suc 𝑛))
5122, 49, 50vtocl 3410 . 2 𝑓𝑛 ∈ ω (𝑓𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)
5221, 51exlimiiv 2011 1 ω ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wex 1852  wcel 2145  wne 2943  wral 3061  Vcvv 3351  wss 3723  c0 4063  {csn 4317  cop 4323   class class class wbr 4787  dom cdm 5250  ran crn 5251  suc csuc 5867  Fun wfun 6024  cfv 6030  ωcom 7215  1𝑜c1o 7709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7099  ax-dc 9473
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-ord 5868  df-on 5869  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-fv 6038  df-1o 7716
This theorem is referenced by:  axdc2lem  9475  axdc3lem  9477  axdc4lem  9482  axcclem  9484
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